Some Examples and Counterexamples on the Relationship Between Conditional and Unconditional Independence

A. García Nogales, P. Pérez Fernández

Some examples and counterexamples are given to illustrate a result included in Nogales and Pérez (2019, to appear in Statistica Neerlandica) on the relationship between conditional and unconditional independence. The mentioned result reads as follows: Let X, Y, Z be three random variables. If X and Y are conditionally independent given Z, then X and Y are independent if and only if are independent the conditional distributions of X and Y given Z. This last condition is equivalent to the uncorrelatedness of the conditional expectations given Z of every pair of bounded real random functions of X and Y. A general discrete framework is built to describe the cited examples and counterexamples.

Keywords: conditional independence Markov kernel

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